What's your favorite computer program?
I've been asking this to some people I know and have found out about some neato stuff. My favorite favorites are Etherpad and Evernote, and those are favorites from Aaron Aiba and Zach Rich. Etherpad allows you to edit a text document on a webpage with other people in real-time, and it color highlights contributed text by user and their color preference. Evernote is a program that is for taking notes that allows you to seamlessly collage frames, crops, links, etc from your screens to an online notebook. Works well as a mobile app too. Thanks to Zach for that one. My favorite computer program is Mathematica 6.0 and it's Manipulate function. The Manipulate function allows you to make sliders for varying parameters of pretty much anything to do with any syntax in Mathematica. Then you can publish these little applets on the web, but it kind of sucks that you have to have Mathematica or download their Live viewer to actually drag the sliders around. But check out http://demonstrations.wolfram.com to see all the beautiful demonstrations. If you don't want to download Mathematica, you can still see everything by clicking "Web Preview" on all the examples and it shows you a little clip of the sliders sliding and the output adjusting. Like, look at this Fucking Robot Snake Arm for as long as you want to before you click "watch web preview" to make it move. Or look at this one to find out how to Buy Watermelons Intelligently. If you're a liberal and/or are under the influence of hallucinogenic, mind-altering chemicals, you might like the Powers of Complex Points. Or if you're conservative, Republican and are scared of the unknown and nontraditional, then you can look at Simulating the 2008 US Presidential Election to remind yourself that you don't have to be scared of the unknown, change, or any combination of frequencies of light in the visual spectrum or lack thereof.

Also, I want to find out Barack Obama and Sarah Silverman's favorite computer programs. Okay, I'm going to go home from work so that I can come back to work this morning for work for when it starts this morning with coffee and on a Friday morning this morning if you will.

On Intelligence by Jeff Hawkins

My review

rating: 4 of 5 stars
Jeff Hawkins does an excellent job of bringing together the story of the neuroscience field from it's historical roots to its current state. He highlights the turns that ended up being the best ones and explains in clear, high-level language an increasingly popular paradigm for a (or possibly the) theory of intelligence. An interesting little voice peeking out from behind this hierarchical prediction paradigm seems to be popping up in other neighborhoods too. If I were to step out on a limb, I'd call the structure of the paradigm fractal-like because of the similarity between different levels of the hierarchy which also can be seen from a view across the hierarchical divisions.

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In Tux's opinion...oh wait, Tux doesn't have an opinion. In general, generalizations sound naïve.

Robots Show That Brain Activity Is Linked To Time As Well As Space

I've always thought that most of brain activity and encoding of its relevant information can probably be abstracted to a system of weighted, probabilistic transmission paths whose signals interfere sort of like normal waveguides. These signals feed back on themselves and and form complex semi-stable states states in the brain.

"...even without explicit spatial hierarchical structure a, functional hierarchy can self-organize through multiple timescales in neural activity."

This is due to the non-linearity of neurons, and similar phenomena occur when non-linear, "non-Newtonian" fluids are given mechanical input.

The Mathematical Theory of Communication by C.E. Shannon

My review

rating: 4 of 5 stars
A humble account by the father of information theory...the first sentence lets you know what you're getting into: "The word communication will be used in a very broad sense to include all of the procedures by which one mind may affect another." I probably wouldn't have read this book if it weren't for the assurance of broadness given from the beginning. It was a quick read, and I was left with the feeling that part of my mind had been tidied up.

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Undecidable Things, Gödel-style!
1. whether an arbitrary Diophantine equation has solutions
2. whether an arbitrary Turing machine will halt
3. validity of a statement in most basic natural number systems if it includes multiplication or division!
4. statements about geometry if angles are allowed!

The Next Awakening
"The next great era of awakening of human intellect may well produce a method of understanding the qualitative contents of equations. Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. Today we cannot see whether Schrödinger's equation contains frogs, musical composers, or morality--or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way." -Richard P. Feynman

Initial Thoughts on Navier-Stokes Equations
I don't think you get real turbulence from solutions to the Navier-Stokes equations from nice initial conditions. Turbulence is chaotic. A system is truly chaotic if it has positive entropy at some time. The Navier-Stokes equations deal with classical fluids, which are fluids that aren't composed of particles. The nature of a classical fluid is a continuous distribution of pressure and a continuous velocity field. This kind of classical fluid, which is the only type that it makes sense to use the Navier-Stokes equations to deal with, has zero entropy and does not exhibit turbulence or chaotic flow. Now, it's accepted that turbulent flow happens when the Reynolds number becomes large, and Reynolds number is the ratio of inertial forces to viscous forces. When fluid is moving slow, viscous forces dominate, and when it's moving fast, inertial forces dominate. In reality, all the fluid we deal with has some entropy because we don't know everything about it (exactly where every piece of it is), but when it's moving slow, we know more. This is because van der Walls forces between the molecules is dominant, so they stay close to one another and pretty much just play music chairs, switching places with one another. We don't have to know the exact state of each piece of fluid to pretty much know an equivalent to one of its future states. So we see flow that is laminar and easy to predict. "Chaos" and turbulence happens when the fluid is moving so fast that the van der Walls aren't dominant and the velocities of the molecules near the edges of the flow are random. Their future states are uncertain. So some of them jump out, and they interact. The patterns they produce are obviously swirly. Imagine one jumping out, and another one hits it in the back and changes its direction a bit. This slows the back-hitter down and causes it to go the other direction a bit. Then the likely thing for the others behind to do is to jump up and hit that one in the back, and you can imagine how a little swirl happens. That swirl bumps into another one. A bunch of swirls get together to make a bigger one and bump into another one that has a completely different shape and all sorts of local asymmetries pop up while the same old story happens on global scales between large numbers of particles. This only happens with particles. You see it in a computer model as well because everything has to be broken up into pieces. The Navier-Stokes equations, however, only deal with infinitesimal pieces...not real ones like molecules. Any chaotic flow or turbulence in real solutions to the Navier-Stokes equations had to be inserted in the initial conditions and it probably wouldn't be considered chaotic flow or turbulence, so I think they only produce "pseudo-chaotic flow". One might call a solution that isn't periodic in time chaotic. For these solutions, there are certain computational attributes that can be ascertained and attributed to either the Navier-Stokes equations themselves, the universe they describe, the being or machine that generated the description of the initial conditions, or a combination of these...probably Turing-completeness.

A Wonderful Passage from Whitehead

from An Introduction to Mathematics by Alfred North Whitehead:

The whole life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining language, they may be termed recurrences of the same event. The rotation of the earth produces the successive days. It is true that each day is different from the preceding days, however abstractly we define the meaning of a day, so as to exclude casual phenomena. Bu with a sufficiently abstract definition of a day, the distinction in properties between two days becomes faint and remote from practical interest; and each day may then be conceived as a recurrence of the phenomenon of one rotation of the earth. Again the path of the earth round the sun leads to the yearly recurrence of the seasons, and imposes another periodicity of the seasons, and imposes another periodicity on all the operations of nature. Another less fundamental periodicity is provided by the phases of the moon. In modern civilized life, with its artificial light, these phases are of slight importance, but in ancient times, in climates where the days are burning and the skies clear, human life was apparently largely influenced by the existence of moonlight. Accordingly or divisions into weeks and months, with their religious associations, have spread over the European races from Syria and Mesopotamia, though independent observances following the moon's phases are found amongst most nations. It is, however, through the tides, and not through its phases of light and darkness, that the moon's periodicity has chiefly influenced the history of the earth.

Our bodily life is essentially periodic. It is dominated by the beatings of the heart, and the recurrence of breathing. The presupposition of periodicity is indeed fundamental to our very conception of life. We cannot imagine a course of nature in which, as events progressed, we should be unable to say: "This has happened before." The whole conception of experience as a guide to conduct would be absent. Men would always find themselves in new situations possessing no substratum of identity with anything in past history. The very means of measuring time as a quantity would be absent. Events might still be recognized as occurring in a series, so that some were earlier and others later. But we now go beyond this bare recognition. We can not only say that three events, A, B, C, occurred in this order so that A came before B, and B before C; but also we can say that the length of time between the occurrences of A and B was twice as long as that between B and C. Now, quantity of time is essentially dependent on observing the number of natural recurrences which has intervened. We may say that the length of time between A and B was so many days, or so many months, or so many years, according to the type of recurrence to which we wish to appeal. Indeed, at the beginning of civilization, these three modes of measuring time were really distinct. It has been one of the first tasks of science among civilized or semi-civilized nations, to fuse them into one coherent measure. The full extent of this task must be grasped. It is necessary to determine, not merely what number of days (e.g. 365.25...) go to some one year, but also previously to determine that the same number of days do go to the successive years. We can imagine a world in which periodicities exist, but such that no two are coherent. In some years there might be 200 days and in others 350. The determination of the broad general consistency of the more important periodicities was the first step in natural science. This consistency arises from no abstract intuitive law of thought; it is merely an observed fact of nature guaranteed by experience. Indeed, so far is it from being a necessary law, that is it not even exactly true. There are divergences in every case. For some instances these divergences are easily observed and are therefore immediately apparent. In other cases it requires the most refined observations and astronomical accuracy to make them apparent. Broadly speaking, all recurrences depend on living beings, such as the beatings of the heart, are subject in comparison with other recurrences to rapid variations. The great stable obvious recurrences--stable in the sense of mutually agreeing with great accuracy--are those depending on the motion of the earth as a whole, and on similar motions of the heavenly bodies.

We therefore assume that these astronomical recurrences mark out equal intervals of time. But how are we to deal with their discrepancies which the refined observations of astronomy detect? Apparently we are reduced to the arbitrary assumption that one or other of these sets of phenomena marks out equal times--e.g. that either all days are of equal length, or that all years are of equal length. This is not so: some assumptions must be made, but the assumption which underlies the whole procedure of the astronomers in determining the measure of time is that the laws of motion are exactly verified. Before this is done, it is interesting to observe that this relegation of the determination of the measure of time to the astronomers arises (as has been said) from the stable consistency of the recurrences with which they deal. If such a superior consistency had been noted among the recurrences characteristic of the human body, we should naturally have looked to the doctors of medicine for the regulation of our clocks.

In considering how the laws of motion come into the matter, note that two inconsistent modes of measuring time will yield different variations of velocity to the same body. For example, suppose we define an hour as one twenty-fourth of a day, and take the case of a train running uniformly for two hours at the rate of twenty miles per hour. Now take a grossly inconsistent measure of time, and suppose that it makes the first hour to be twice as long as the second hour. Then, according to this other measure of duration, the time of the train's run is divided into two parts, during each of which it has traversed the same distance, namely, twenty miles; but the duration of the first part is twice as long as that of the second part. Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period. Thus the question as to whether the train has been running uniformly or not entirely depends on the standard of time which we adopt.

Now, for all ordinary purposes of life on the earth, the various astronomical recurrences may be looked on as absolutely consistent and furthermore assuming their consistency, and thereby assuming the velocities and changes of velocities possessed by bodies, we find that the laws of motion, which have been considered above, are almost exactly verified. But only almost exactly when we come to some of the astronomical phenomena. We find, however, that by assuming slightly different velocities for the rotations and motions of the planets and stars, the laws would be exactly verified. The assumption is then made and we have, in fact thereby, adopted a measure of time, which is indeed defined by reference to the astronomical phenomena, but not so as to be consistent with the uniformity of any one of them. But the broad fact remains that the uniform flow of time on which so much is based is itself dependent on the observation of periodic events.

A Fundamental Thought
"If a function x(t) contains no frequencies higher than B cycles per second, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart." -Claude Shannon

It's a good thing.